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Advanced Geometry : How to graph an exponential function

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Example Question #2 : Graph A Polynomial Function

Give the -intercept of the graph of the function

$f(x) = 5 \cdot 4^{x- 3}- 3$

Round to the nearest tenth, if applicable.

Possible Answers:

( -2.92, 0)

(2.63,0)

(3.65,0)

( -3.08, 0)

The graph has no -interceptx

Correct answer:

(2.63,0)

Explanation:

The -intercept is (a,0) , where f(a)= 0 :

$f(x) = 5 \cdot 4^{x- 3}- 3$

$5 \cdot 4^{a- 3}- 3 = 0$

$5 \cdot 4^{a- 3}- 3 + 3= 0 + 3$

$5 \cdot 4^{a- 3}= 3$

$5 \cdot 4^{a- 3} \div 5= 3 \div 5$

$4^{a- 3} = 0.6$

$\ln \left (4^{a- 3} \right )=\ln 0.6$

$(a- 3)\ln 4 =\ln 0.6$

$a- 3=\frac{\ln 0.6}{\ln 4 } \approx \frac{-0.5108}{1.3863}\approx -0.3685$

$a- 3+3 \approx -0.3685 + 3$

$a \approx 2.63$

The -intercept is (2.63,0) .

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Example Question #1 : How To Graph An Exponential Function

Give the -intercept of the graph of the function

$f(x) = 5 \cdot 4^{x- 3}- 3$

Round to the nearest hundredth, if applicable.

Possible Answers:

(0, -2.92)

(0, -3.08)

The graph has no -intercept

( 0, 2.63)

(0,3.65)

Correct answer:

(0, -2.92)

Explanation:

The -intercept is (0, f(0)) :

$f(x) = 5 \cdot 4^{x- 3}- 3$

$f(0) = 5 \cdot 4^{0- 3}- 3$

$f(0) = 5 \cdot 4^{ - 3}- 3$

$f(0) = 5 \cdot \frac{1}{64}- 3$

$f(0) = \frac{5}{64}- 3$

$f(0) \approx 0.08 - 3 \approx -2.92$

(0, -2.92) is the -intercept.

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Example Question #1 : How To Graph An Exponential Function

Give the vertical asymptote of the graph of the function

$g(x) = 16 \cdot 4^{x}- 3$

Possible Answers:

y = -2

The graph of has no vertical asymptote.

x = -3

x = 3

y = 2

Correct answer:

The graph of has no vertical asymptote.

Explanation:

Since 4 can be raised to the power of any real number, the domain of is the set of all real numbers. Therefore, there is no vertical asymptote of the graph of .

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Example Question #3 : How To Graph An Exponential Function

Give the horizontal asymptote of the graph of the function

$g(x) = 9 \cdot 3^{x}- 3$

Possible Answers:

y = 2

The graph has no horizontal asymptote.

y = 3

y = -3

y = -2

Correct answer:

y = -3

Explanation:

We can rewrite this as follows:

$g(x) = 9 \cdot 3^{x}- 3$

$g(x) = 3^{2} \cdot 3^{x}- 3$

$g(x) = 3^{x+2}- 3$

This is a translation of the graph of $f(x) = 3^{x}$ , which has y = 0 as its horizontal asymptote, to the right two units and down three units. Because of the latter translation, the horizontal asymptote is y = -3 .

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Example Question #1 : How To Graph An Exponential Function

If the functions

$f(x) = e^{x}+ 9$

$g(x)= 4e^{x}- 7$

were graphed on the same coordinate axes, what would be the -coordinate of their point of intersection?

Round to the nearest tenth, if applicable.

Possible Answers:

The graphs of and would not intersect.

2.7

14.3

1.7

5.3

Correct answer:

14.3

Explanation:

We can rewrite the statements using for both f(x) and g(x) as follows:

$y = e^{x}+ 9$

$y = 4e^{x}- 7$

To solve this, we can multiply the first equation by , then add:

$-4y = -4e^{x}-36$

$\underline{y = 4e^{x} \; \; - 7}$

-3y =

$-3y \div (-3) = -43 \div (-3 )$

$y \approx 14.3$

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Example Question #5 : How To Graph An Exponential Function

If the functions

$f(x) = e^{x}+ 9$

$g(x) = 4e^{x}- 7$

were graphed on the same coordinate axes, what would be the -coordinate of their point of intersection?

Round to the nearest tenth, if applicable.

Possible Answers:

The graphs of and would not intersect.

1.7

14.3

2.7

5.3

Correct answer:

1.7

Explanation:

We can rewrite the statements using for both f(x) and g(x) as follows:

$y = e^{x}+ 9$

$y = 4e^{x}- 7$

To solve this, we can set the expressions equal, as follows:

$4e^{x}- 7 = e^{x}+ 9$

$4e^{x}- 7 - e^{x} + 7 = e^{x}+ 9 - e^{x} + 7$

$3e^{x} = 16$

$3e^{x}\div 3 = 16 \div 3$

$e^{x} = \approc 5.3333$

$x \approx \ln 5.333 \approx 1.7$

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Example Question #131 : Advanced Geometry

Find the range for,

$y=3^{x}-2.$

Possible Answers:

y< -2

$y\leq -2$

y>-2

$y\geq -2$

Correct answer:

y>-2

Explanation:

$The\ graph\ of\ the\ function\ y=3^{x}-2\ has\ an\ asymptote\ at\ y=-2.$

$An\ asymptote\ on\ a\ graph\ is\ a\ value\ that\ a \ function\ approaches\ but\ never\ touches.$

$Since\ this\ graph\ approaches\ but\ never\ touches\ -2\ we\ use\ the >\ symbol.$

$\\Also\ because\ the\ 3\ is\ positive\ that\ means\ the\ graph\ is\ concave\ up\ so\ >\ is\ chosen\ instead\ of\ <.$

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Example Question #131 : Advanced Geometry

An important part of graphing an exponential function is to find its -intercept and concavity.

Find the -intercept for

$y=2(3)^{x}-5$

and determine if the graph is concave up or concave down.

Possible Answers:

$y-int: (0,-5); Graph\ is\ concave\ up.$

$y-int: (0,-3); Graph\ is\ concave\ up.$

$y-int: (0,-5); Graph\ is\ concave\ down.$

$y-int: (0,-3); Graph\ is\ concave\ down.$

Correct answer:

$y-int: (0,-3); Graph\ is\ concave\ up.$

Explanation:

$\\The\ concavity\ of\ an\ exponential\ graph\ can\ be\ determined\ by\ its\ a-value.$

$This\ function\ is\ in\ the\ form\ y=a(b)^{x}+k.$

$\ So\ the\ a\ value\ is\ 2, based\ on\ the\ function\ we\ were\ given\ y=2(3)^{x}-5$

$If\ the\ a\ value\ is\ positive,\ the\ graph\ will\ be\ concave\ up$ .

$If\ the\ a\ value\ is\ negative,\ the\ graph\ will\ be\ concave\ down.$

$To\ find\ the\ y-intercept,\ plug\ in\ x=0, and\ keep\ in\ mind\ order\ of\ operations,$

$\ Exponent first, then\ multiply.$

$y=2(3)^{0}-5$

y=2(1)-5=-3

$The\ y-intercept\ is\ (0,-3).$

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Example Question #41 : Graphing

Give the equation of the vertical asymptote of the graph of the equation $f(x) = 2e^{x-3}+ 7$ .

Possible Answers:

x= 3

x= 2

x= 7

The graph of f(x) has no vertical asymptote.

x= e

Correct answer:

The graph of f(x) has no vertical asymptote.

Explanation:

Define $g(x) = e^{x}$ . In terms of g(x) , f(x) can be restated as

f(x) = 2g( x-3 )+ 7

The graph of f(x) is a transformation of that of g(x) . As an exponential function, g(x) has a graph that has no vertical asymptote, as g(x) is defined for all real values of ; it follows that being a transformation of this function, f(x) also has a graph with no vertical asymptote as well.

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Example Question #131 : Advanced Geometry

Give the equation of the horizontal asymptote of the graph of the equation $f(x) = 2e^{x-3}- 7$ .

Possible Answers:

y = 2

y = 3

y = -7

y = e

The graph of f(x) has no horizontal asymptote.

Correct answer:

y = -7

Explanation:

Define $g(x) = e ^{x}$ . In terms of g(x) , f(x) can be restated as

$f(x) = 2e^{x-3}- 7$ .

The graph of g(x) has as its horizontal asymptote the line of the equation y = 0 . The graph of f(x) is a transformation of that of g(x) - a right shift of 3 units ( x-3 ), a vertical stretch ( ), and a downward shift of 7 units ( ). The right shift and the vertical stretch do not affect the position of the horizontal asymptote, but the downward shift moves the asymptote to the line of the equation y = -7 . This is the correct response.

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Advanced Geometry : How to graph an exponential function

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #2 : Graph A Polynomial Function

Example Question #1 : How To Graph An Exponential Function

Example Question #1 : How To Graph An Exponential Function

Example Question #3 : How To Graph An Exponential Function

Example Question #1 : How To Graph An Exponential Function

Example Question #5 : How To Graph An Exponential Function

Example Question #131 : Advanced Geometry

Example Question #131 : Advanced Geometry

Example Question #41 : Graphing

Example Question #131 : Advanced Geometry

All Advanced Geometry Resources

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